Pentadiagonal

A pentadiagonal matrix is a square matrix in which all entries are zero except on five diagonals: the main diagonal, the two diagonals just above the main, and the two diagonals just below the main. Equivalently, an n×n pentadiagonal matrix has nonzero entries on the diagonals i, i±1, and i±2, for as many rows and columns as allowed by boundary conditions. It is a type of banded matrix with a bandwidth of 2 on each side.

Pentadiagonal matrices commonly arise in numerical linear algebra when discretizing certain differential equations, especially fourth-order problems, or when applying high-order finite difference schemes. They also appear in some discretizations of beam and plate bending problems.

Storage and computation: Because only five diagonals may be nonzero, these matrices require O(n) storage. Specialized algorithms exist for solving pentadiagonal systems that exploit the banded structure, delivering efficient LU-like factorizations that preserve most of the pentadiagonal form. In general, ordinary LU factorization with pivoting can destroy the five-diagonal structure, so solvers often implement structure-preserving methods. The computational cost is linear in the matrix size with small constants relative to dense methods.

Boundary conditions affect the first and last two rows, which may have fewer nonzero entries. Inverse of a pentadiagonal matrix is not necessarily pentadiagonal. See also: tridiagonal matrix, banded matrix, and pentadiagonal solvers.